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Definition: "Lies on" Relation

Let $\mathcal P$, $\mathcal L$ and $\Pi$ be sets of points, straight lines, and planes respectively.

The “lies on” relation $R$ is an arbitrary relation $R\subseteq \mathcal P\times\mathcal L.$ We say that a point $A\in \mathcal P$ lies on a straight line $h\in \mathcal L$ if the ordered pair $(A,h)$ is contained in $R$ in set-theoretical sense, i.e. $(A,h)\in R.$

Similarly, the “lies on” relation $S\subseteq \mathcal P\times\Pi$ means that a point $A\in \mathcal P$ lies on a plane $\alpha \in \Pi$ if the ordered pair $(A,\alpha)\in S$ in set-theoretical sense.


| | | | | created: 2019-12-21 05:28:01 | modified: 2019-12-21 06:36:00 | by: bookofproofs | references: [6260], [8231], [8251], [8324]

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Bibliography (further reading)

[8231] Berchtold, Florian: “Geometrie”, Springer Spektrum, 2017

[8324] Hilbert, David: “Grundlagen der Geometrie”, Leipzig, B.G. Teubner, 1903

[8251] Klotzek, B.: “Geometrie”, Studienb├╝cherei, 1971

[6260] Lee, John M.: “Axiomatic Geometry”, AMC, 2013